Remarks on scope ambiguity*
Dag Westerståhl
Department of Philosophy, Göteborg University
Abstract
It has been argued that some scope ambiguities are not really structural:
the underlying phrase structure is ambiguous, and no further structure can
be independently motivated. Assuming this, with a simple case of quantifier
scoping as illustration, we look at the consequences for semantics, and in
particular for the principle of compositionality. I observe that attempts to
save the usual version of compositionality by revising the notion of
meaning (in particular by taking the new meaning to be the set of old
meanings) are not guaranteed to work. My main claim is that if one accepts
instead a relational semantics, where structured as well as lexical
*
Jens Allwood is a linguist with very broad philosophical interests. We have been
colleagues for an inordinate number of years, and still meet regularly to talk about
linguistic and philosophical matters. In the context of the present paper, I recall that
Jens was one of the first to introduce Montague Grammar in Sweden, in the 70’s. I think
he would agree with my claim that Montague’s treatment of scope ambiguity is not
wholly convincing. Since those early days, however, our approaches to our common
interest – language – have increasingly diverged, and I am afraid he might not agree
with much else I say here. But then, Jens rarely agrees with much of what anyone says.
expressions are allowed to have more than one meaning, the compositionality principle can be adapted without loss of explanatory power.
Keywords: scope, ambiguity, compositionality, relational semantics
1. Introduction
Since Montague we have grown accustomed to the idea that structural
ambiguities, and in particular ambiguities of scope, always have a syntactic
counterpart. To every syntactic rule α corresponds a unique semantic rule
rα, and since there is nothing more to semantics than rules of this form and
the input from the lexicon, all meaning variation in a surface string goes
back to either lexical ambiguity or differences in the syntactic derivation of
the string. Lexical ambiguity can be dealt with by simply introducing
different terms for different meanings of a lexical item, and so all
ambiguities are nicely disposed of.
This is an elegant picture, and it was no trivial matter at the time to see that
it could be rigorously carried through for large parts of natural languages.
Today we should perhaps be less impressed by this particular aspect of
Montague’s achievement; it has become almost obvious that if one wants to
account for ambiguity in this way, it is possible to do it. So what are the
reasons for choosing this or that syntactic format? Specifically, if the only
reason for introducing certain syntactic rules is to account for semantic
ambiguity, perhaps that reason is not good enough?
2
In this short and programmatic paper I shall not discuss syntax, but only
indicate, by means of a simple example, some consequences of abandoning
Montague’s idea and allowing syntax to ‘underdetermine’ semantics, even
as regards some issues of scope. Such a move is rather natural and has been
suggested many times in the literature. But what has not been discussed
with equal care, I think, is the methodological consequences that ensue. In
particular, how does this move affect another basic principle of semantics,
that of compositionality?
2. The example
The example is a basic case of quantifier scope ambiguity, but rather than
the philosopher’s standard “Every man loves a woman” I choose a slightly
more interesting (and realistic) version. Consider the following two
sentences:
(1) Two countries objected to every proposal.
(2) Two knives belonged to every equipment.
I will make a number of assumptions concerning (1) and (2). The first two
should be completely uncontroversial.
(a) To understand what these sentences say (e.g. to decide whether they are
true or not) one needs a context: an utterance situation, some worldly
background, etc. However, ‘default’ contexts easily spring to mind,
allowing one to rather effortlessly make sense of the sentences. For
example, (1) could be about an assembly of representatives of nations, such
3
as UN or EU, discussing measures to adopt on some sensitive matter. And
(2) could be about the equipments issued to a group of boy scouts before
one of their hikes. (Think of (2) uttered after the hike by the leader of the
(say) 10 boys, who goes on to say: “But so far only 19 knives have been
returned!”)
(b) In these ‘default’ contexts, the determiners two and every have different
scope in (1) and (2). (1) (normally) says that there were two particular
countries whose representatives opposed every proposal made during the
assembly, not that for every such proposal there were two countries
(possibly different ones for different proposals) opposing it. But (2)
(normally) does not say that there were two specific knives that somehow
belonged to each of the equipments. Rather, it (normally) says that each
equipment contained two knives (most likely different knives for different
equipments).1
I think (a) and (b) are perfectly obvious. Nevertheless, it is sometimes
argued that quantifier scope ambiguities are rare and that the surface syntax
provides a default reading. (1) and (2) should make one suspicious of such
claims.2 But that suspicion needs to assume something about the surface
syntax. I think (almost) no one would object to the following:
(c) (1) and (2) have the same surface syntax.
1
Furthermore, it is easy to imagine a context where (1) has the other, ’inverse scope’
reading. That reading is the default for (2) but, (with greater effort) one can imagine a
situation where the other reading was intended instead.
2
Note that the claim does not follow in any straightforward way from the (in itself
interesting) fact that speakers normally and without thought pick out one of the scopes
in actual utterances. The issue is not the ambiguity of utterances but of the sentences
they use.
4
The next assumption might look more debatable, but it really isn’t, since it
is meant to be independent of any particular format for syntax. Pick your
favorite syntactic framework: the odds are overwhelming that (a suitably
adapted version of) (d) below will hold:
(d) The basic surface structure of both (1) and (2) can be represented as
follows.
(T)
The labels here have no importance; nor does it matter which phrases are
taken to be ‘heads’ and which are ‘specifiers’, nor whether additional (Xbar style) labels are inserted. We are only talking about basic structure.
Now, we all know how to endow T with additional structure in ways that
give (1) and (2) different syntactic analyses, corresponding to the different
scopes of the determiners. For example, we may allow transformations
(movement, quantifier raising) to turn T into one of two more complex
trees (logical forms) where scope is fixed, and argue that meaning is only
assigned to these latter forms. Or we may have syntactic rules that generate
the more complex forms directly, without using T, or perhaps using T for
5
one of the two scopes (the default scope of (1)), and a more complex form
for the other.
My last assumption, (e) below, is the only controversial one in the present
context. It seems to me (and it has seemed to others, for example, Pelletier
1999) that the only reason to introduce such additional structure is to
account for the ambiguity. In terms of syntax alone, T is all the structure
there is. That is, in this case there is no syntactic structure beyond phrase
structure:
(e) The syntactic structure of (1) and (2) is (essentially) T.
I think one can make a pretty good case for (e), but this will not be pursued
here. The point is rather the following: Assuming (a) – (e), what are the
consequences?
3. Consequences for compositionality
The principle of compositionality says that the meaning of a complex
expression is determined by the meanings of its parts and the ‘mode of
composition’ (usually, the syntactic rule used to derive the expression from
its immediate parts). This presupposes that each expression has (at most)
one meaning, or, as Montague put it, that expressions have been
disambiguated. Two ways of obtaining such disambiguation were sketched
above, and in both cases classical compositionality can be maintained.3
3
Barker and Jacobson (2007) call the first option (which is consonant with generative
grammar, for example in many of the forms it has been given by Chomsky during the
6
However, if our example above is to be taken seriously, it may seem that
we simply have to give up compositionality. Pelletier (1999) draws this
conclusion, taking the occurrence of non-lexical ambiguities not
explainable by differences in syntactic derivation as evidence that the
principle of compositionality fails. But there are at least two other options
available.
The first is to revise the notion of meaning. Standardly, the meaning of a
sentence is its truth conditions, often taken to be embodied in a proposition
recording its truth value under various circumstances or possible worlds.
Analogously, the meanings of parts of sentences are entities that suitably
contribute to propositions. One idea, then, is to take meanings to be
something less specific, so that when the semantic analysis is done, the
meaning of the whole sentence is not yet a proposition but something more
abstract, which yields a proposition by some further operation, perhaps a
pragmatic one. Applying the same idea to parts of sentences, each syntactic
unit will still have a unique meaning. Of course, such a maneuver does not
guarantee compositionality, but it removes one obstacle to it.
In fact, this move is quite familiar, although the reasons for making it have
usually not been tied to a wish to uphold compositionality (indeed, the
move has sometimes been tied to a rejection of compositionality). Instead,
years) indirect compositionality, since transformations are syntactic rules without
semantic counterparts. The second option on the other hand (which is consonant with
Montague Grammar), is direct. This issue does not really concern compositionality but
rather what computer scientists have called the Domain Principle, i.e. that subexpressions of meaningful expressions are also meaningful. Hodges (2001) notes that
compositionality has natural formulations in either case (equivalent under DP; see
section 8 below). Dowty (2007) points out that the distinction is not absolute, since an
indirect semantics can often be recast as a direct one.
7
the motivation has been, roughly, that semantics should not just give us the
means to derive all possible propositions expressed by a sentence, but also
say something about how speakers usually arrive at one of these
propositions. For Montague, this belonged to pragmatics, but the
semantics-pragmatics borderline is something that has been much debated
recently. However the line is drawn, a marked tendency has been that
semantics should become more relevant to actual understanding and
communication than Montague’s framework allowed.
I will have something brief to say about this kind of move and its relation
to compositionality in section 6, but my main interest here is in the second
option referred to above. This idea is simply to give up, not
compositionality, but the assumption that every expression has a unique
meaning. That is, the idea, which is rarely carried out in the literature, is to
let meaning association be a relation rather than a function, and then to
attempt a reformulation of the compositionality principle for relational
semantics.4 What such an attempt might look like will be sketched in
section 8.
4. Why compositionality?
At this point the reader might wonder why I am making such a fuss about
compositionality. Why do I seek to uphold compositionality in the
framework of a relational semantics, and why am I worried (see section 6)
4
One of the rare exceptions is Debusmann et al. (2004), who give a relational grammar
in this sense for a fragment of English (in a Dependency Grammar format), and for
partly the same reasons as here. They do not discuss compositionality, however. Thanks
to Joakim Nivre for bringing this work to my attention.
8
that those who advance the first option mentioned above are not
sufficiently concerned with upholding it?
My answer here must be brief. I am simply adhering to a tradition which
can be said to begin with Frege, and which is largely adhered to among
linguists and philosophers of language today, according to which there are
certain remarkable facts about human language that need to be explained,
and compositionality is an important ingredient in the explanation. These
facts are, roughly, that speakers of a language are able to effortlessly
understand sentences they have never heard or come across before,
sentences that moreover express thoughts (propositions) that are likewise
new to the speaker. This is often called the productive aspect of language
use. It should be contrasted with what can be called its creative aspect. The
latter concerns speakers’ ability to invent new linguistic constructions
(fresh idioms and metaphors are examples), thus creating new semantic
rules. It is remarkable too, I think, that speakers are able to do this with
such apparent ease, and that others are often able to correctly ‘guess’ (if
that is what they do) how these constructions and rules are to be
understood. But productivity does not rely on creativity. On the contrary,
its explanation in terms of compositionality is precisely that the given
(syntactic and semantic) rules of language are used, and that it is the
knowledge of these rules, together with lexical knowledge, which allows
the speaker to immediately grasp applications of these same rules in ways
she has not done or come across before, although they are implicit in the
rules.5
5
The number of ways in which these rules can be applied is infinite, or at least very
large (even with a small lexicon), much larger than the instances a speaker actually
9
It must be added that all of the above has been denied. Philosophers and (to
a lesser extent) linguists and logicians have denied that Frege expressed or
believed in compositionality, or that the linguistic facts referred to exist, or
that they are remarkable or in need of explanation, or that compositionality
can explain them or indeed explain anything at all. But this is not the place
to review these arguments. The position I am taking is not deviant, indeed
it is mainstream, and here I will simply presuppose that there is a value in
maintaining compositionality.6
5. Compositionality and context
Before coming to the point a word must be said about context. That natural
language is strongly context-dependent is a familiar fact, though recently
there has been much debate about how this is to be accounted for, whether
it belongs to semantics or pragmatics, etc. Most of the people involved
agree at least that certain basic indexicals – like I, you, here, tomorrow –
are context-dependent in a very systematic way, largely determined by
linguistic rules. It is thus reasonable to incorporate such contextdependence in the corresponding meanings (for example, by letting the new
meanings be functions from contexts to the old meanings). But doing that
has consequences for compositionality. Roughly, there are now additional
ways that the compositionality of a proposed semantics might fail: not only
may substitution of synonymous parts fail to preserve meaning, but also
comes across during her lifetime. Note, furthermore, that I am not saying that
compositionality could not play a role in explaining linguistic creativity too.
6
For recent overviews of this debate, see Szabó (2007) or Pagin and Westerståhl (to
appear).
10
what Pagin and Pelletier (to appear) call context shift may have that effect.
Put differently, there are distinct versions of the principle of
compositionality, depending on whether one composes meanings before or
after the contextual contributions are in place.
But I mention this only to disregard it. Here I want to focus on just one
particular phenomenon – how certain kinds of ambiguity affect
compositionality – and therefore I omit semantic features not immediately
relevant to this issue. It is possible to bring out the effects of ambiguity
without invoking context-dependence. However, I want to emphasize that
in a fuller treatment, context-dependence should not be disregarded. In fact,
context-dependence and ambiguity are related in interesting ways that need
to be accounted for, but that must be left for another occasion.
6. Set semantics and other forms of underdetermined meanings
As I said, one way to restore compositionality in the face of ambiguity is to
revise the notion of meaning. Perhaps the simplest idea here is to use the
set of possible meanings instead. Suppose meaning assignment is a relation
R between (possibly analyzed) expressions and meanings, rather than a
single-valued function from expressions to meanings. For each expression
e, let
Re = {m: R(e,m)}
11
be the set of meanings of e. (Assume a set E of expressions and a set M of
meanings to be given.) Now introduce a new semantics µR by defining, for
each e in E,
µR(e) = Re
This semantics is single-valued – each expression has a unique meaning –
and so the issue of its compositionality may be raised in the standard way.
This way to handle ambiguity is quite common. For example, Pelletier
(1999) indicates that lexical ambiguity can be treated by giving each lexical
item the set of its various meanings as semantic value (new meaning), and
these values are then ‘passed up’ to complex expressions in a natural way.
Likewise, Cooper storage (see Cooper 1983) can be seen as a more
sophisticated version of the same idea, this time not for lexical items but
for
ambiguities
of
scope.
Furthermore,
an
intuitively
appealing
propositional interpretation of Re, in the special case when e is a sentence,
is simply as the disjunction of its members.
A lot can be said about this idea and its manifestations in the literature, but
here I will make just one comment: In general, there is no principled
guarantee that the set semantics is compositional, even when the original
semantics is.
This may seem puzzling. Suppose a binary syntactic operation α combines
ambiguous lexical items a and b, and that a corresponding semantic
operation rα combines their meanings. Can’t we then just define a new
12
semantic operation that operates on the sets of meanings of a and b,
respectively, and yields the set of all rα(m1,m2), where m1 is a meaning of a
and m2 is a meaning of b (and rα(m1,m2) is defined), as output? The
problem is that there may be all kinds of constraints on when the
combination is meaningful. The simple-minded ‘lifting’ of rα just indicated
may give a (set) meaning to the combination when it shouldn’t, and if we
are unlucky, this can result in an irreparable failure of compositionality. For
example, it could happen that α(a,b) is meaningful but α(a,c) isn’t, or that
one of the meanings of α(a,b) is not available for α(a,c), even though b and
c have the same meanings. If so, our ‘lifting’, and indeed any set semantics,
will give α(a,b) and α(a,c) different meanings, violating compositionality.7
This example is abstract but the point, I think, is general. Sets of ‘old’
meanings are rather different objects than the ‘old’ meanings themselves,
and it is not obvious that natural operations on ‘old’ meanings
straightforwardly lift to equally natural operations on sets, or indeed to any
operations on sets. My point, therefore, is that although switching to set
semantics (or its variants) looks like a natural way to get around ambiguity,
the issue of compositionality is not thereby automatically resolved; on the
contrary it needs to be approached anew.
7
The counterexample – which is similar to one given in van Deemter (1996) – involves
failure of what Hodges (2001) calls the Husserl property, roughly that intersubstitution
of synonymous expressions preserves meaningfulness. It is debatable whether natural
languages have this property. In any case, it is not hard to find natural proposals for
semantics that violate it.
13
7. Relational semantics
We now come to what (to me) looks like the most straightforward way to
deal with ambiguity, especially ambiguity that is neither lexical (or induced
by lexical ambiguity) nor structural, as in our example from section 3.8 The
idea is: Accept that the semantics (the meaning assignment) is relational!
Prima facie, the fact that some sentences, even when their syntactic
structure is taken into account, have more than one meaning doesn’t seem
to be a great obstacle to understanding and communication. The hearer
needs to pick up the reading that the speaker intended, but this could be
achieved in a number of ways. It is not at all clear that imposing a more
complex underlying structure is especially helpful. After all, with lexical
ambiguity, as in John went to the bank (when both a river bank and a
financial bank are ‘salient’ in the utterance situation), the hearer has to rely
on other cues to get at the right interpretation, and there is no reason, it
seems to me, that the same could not hold for quantifier scope ambiguity.
Our question, then, is whether such a relational semantics destroys the idea
of compositionality, or the notion that compositionality is a key ingredient
in an explanation of how language works.
8
Of course there are ambiguities that are obviously structural, like the one in Old men
and women were asked to get in the lifeboat. The assumption here is just that some nonlexical ambiguities are not structural, with the different scopes of (1) and (2) in section
3 as paradigmatic examples.
14
8. Compositionality in a relational framework
To answer the question just posed it is necessary to separate two
formulations of compositionality, which are (almost) indistinguishable in
the single-valued case. I call them the Rule version and the Substitution
version, and only give approximate and non-technical formulations here;
this suffices to illustrate the main points. First, consider the single-valued
case:
(Rule)
For every syntactic rule rsyn there is a corresponding semantic
operation rsem such that the meaning of the result of applying
rsyn to certain expressions is the result of applying rsem to the
meanings of those expressions.
That is, the meaning of the whole is determined by the meanings of the
parts, and the mode of composition (rsyn).
(Subst)
Replacement of (not necessarily immediate) sub-expressions of
a complex expression by synonymous expressions results – if
the result is meaningful at all – in a complex expression
synonymous with the original one.
That is, replacement of synonymous parts (when allowed) preserves
meaning. It is not hard to give precise versions of (Rule) and (Subst), and
to show that, provided sub-expressions of meaningful expressions are
always meaningful (which (Rule) but not (Subst) presupposes; cf. footnote
3), they are equivalent (see Hodges 2001 or Westerståhl 2004).
15
(Rule) presupposes a single-valued meaning assignment: witness phrases
like “the meaning of”. (Subst) does so too, in a less perspicuous way, in
that it uses a straightforward notion of synonymy: two expressions are
synonymous if (both are meaningful and) they have the same meaning.
Clearly it is the (Rule) formulation that best corresponds to the intuitive
motivation behind compositionality: it offers an explanation of linguistic
productivity in terms of knowledge of syntactic rules and their semantic
counterparts. Roughly, the first explains why we recognize new sentences
as well-formed, and the second why we effortlessly see what they mean.9
And it is this formulation that readily extends to the case of a relational
semantics, where an expression may have several meanings:
(RuleAmb) For every syntactic rule rsyn there are corresponding semantic
operations r1,sem,…, rk,sem such that the meanings of the result of
applying rsyn to certain expressions is the result of applying
ri,sem, for some i, to meanings of those expressions.
This formulation is still imprecise, and there are in fact some niceties
pertaining to finding an exact version of (RuleAmb). I will not go into them
here. But I think the idea is clear: For each syntactic rule there is now
(possibly) more than one corresponding semantic rule. Thus, finding a
meaning of the complex expression involves having access to some
meanings of its immediate constituents, and applying one of the
corresponding semantic operations to those meanings. One must choose, as
it were, meanings of the parts, and also the semantic operation to apply.
9
Of course, much more is required for such an explanation to work, in particular that
the semantic operations are easy (for us) to compute, but that is another issue.
16
This is a little more complex than in the case when there are unique
meanings of the parts and just one operation to apply, but not much. After
all, already with lexical ambiguity one has to choose among the possible
meanings of lexical items; the only new element here is that one also has to
choose a rule. But this seems equally feasible; all the available rules are
specified in advance, and depend only on the syntactic ‘mode of
combination’, just as before.
Consider our sample sentences (1) and (2) in section 3. The syntactic
structure of both is the tree T. With a relational semantics, and assuming
there are no lexical ambiguities, each sentence has two meanings,
corresponding to the two possible ‘orders’ between the quantifiers.
Suppose ambiguity sets in only at the last rule application, the one that
forms the sentence from the NP and the VP (other accounts are also
possible). At this stage there are two semantic rules: one for each scoping.
We saw that for these two sentences, the natural reading involves one
scoping for (1) and the other one for (2). This choice can clearly be
described as a choice between one of two semantic rules, both of which are
given, and ‘known’ by the speaker, in advance.
I end with three comments, stated here without proof. They can be proved
rigorously in a suitably precise setting.
First, the precise version of (RuleAmb) is such that, when the relational
semantics happens to be single-valued, (RuleAmb) is equivalent to the usual
(Rule).
17
Second, even when n = 1 in (RuleAmb), so that each syntactic rule
corresponds to exactly one semantic operation, there is still room for
lexical ambiguity. One can show that (RuleAmb) in this case gives a
reasonable account of how lexical ambiguity is propagated to complex
expressions. More generally, (RuleAmb) is intended to work in the presence
of all three kinds of ambiguity: lexical, structural, and the third kind
exemplified in section 3.
Third, a difference with the single-valued case, however, is that there
seems to be no obvious substitution version of (RuleAmb). The reason, I
think, is that there is no obvious relation of synonymy in relational
semantics. There are several natural synonymy relations: the strongest one
saying that two expressions are synonymous if they have exactly the same
meanings. (This is the synonymy corresponding to the set semantics of
section 6.) But there are also natural weaker versions. It can be shown that
none of these is equivalent to (RuleAmb). Apparently, with a relational
semantics, the notion of synonymy becomes more problematic.
9. Conclusion
I have argued that even if we give up the idea that each (structured)
expression has a unique meaning, the idea of compositionality remains, and
so does the explanation of the ‘remarkable facts’ about human language
that Frege and many others have noted. The main part of the argument was
an indication of how to formulate the principle of compositionality within a
relational semantics, a formulation that extends the standard version for
functional semantics. Clearly, for the argument to go through one needs a
18
precise version of this principle, and an investigation of its properties, just
as the usual version has been investigated and is now quite well
understood. Also, one needs to look at a variety of applications to concrete
cases. These are things I hope to come back to in the future.
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